Scenario-Free Stochastic Programming

Transcription

1 Scenario-Free Stochastic Programming Wolfram Wiesemann, Angelos Georghiou, and Daniel Kuhn Department of Computing Imperial College London London SW7 2AZ, United Kingdom December 17, 2010

2 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

3 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

4 Deterministic Optimisation Deterministic optimisation model: minimise x f(x,ξ) subject to g(x,ξ) 0, where x are decision variables ξ are (precisely known) parameters Real world is uncertain. Why not use ξ = E [ ξ ]?

5 Deterministic Optimisation Image Source: Sam L. Savage, Stanford University alive(e[position]) = true, but E[alive (position)] = false!

6 Portfolio optimisation: Deterministic Optimisation wealth(e[stock returns]) vs. E[wealth(stock returns)]

7 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

8 Two-stage stochastic program: minimise x,y Stochastic Programming subject to x X, f(x)+e[q(y(ξ); x,ξ)] y(ξ) Y(x,ξ) P-a.s. x X y(ξ) Y(x,ξ) ξ f(x) q(y(ξ); x,ξ)

9 Stochastic Programming Multi-stage stochastic program: several recourse decisions capacity expansion: several investment stages production planning: annual production plan (seasonalities) portfolio optimisation: rebalancing, asset & liability mgmt.... x y 1 (ξ 1 ) y 2 (ξ 1,ξ 2 )... ξ 1 ξ 2 f(x) q 1 (y 1 (ξ 1 ); x,ξ 1 ) q 2 (y 2 (ξ 1,ξ 2 ); x,ξ 1,ξ 2 )

10 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

11 Discretise distribution: into scenarios Scenario-Based S.P. minimise x,y f(x)+ s S p s q(y s ; x,ξ s ) subject to x X, y s Y(x,ξ s ) s S. y(ξ 1 ) x X y(ξ 2 ) y(ξ 2 ) f(x) y(ξ 3 ) ξ q(y(ξ 2 ); x,ξ 2 )

12 Scenario-Based S.P. Portfolio optimisation: today tomorrow

13 Multi-stage stochastic program: Scenario-Based S.P. Solution time:

14 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

15 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

16 Linear Decision Rules 1 Step 1: linearise decision rule minimise x,y, y subject to x X, f(x)+e[q(yξ + y; x,ξ)] Yξ + y Y(x,ξ) P-a.s. Yξ Y 2 Y 1 ξ 2 ξ 1 x X Yξ + y Y(x,ξ) ξ f(x) q(yξ + y; x,ξ) 1 Ben-Tal et al., Math. Programming, 2004.

17 Linear Decision Rules Step 2: reformulate semi-infinite P-a.s. constraint for Y(x,ξ) = {y(ξ) : y(ξ) [0, 5]}: } y 0 10[ Y 1 ] [ Y 2 ] + y [Y 1 ] + 10[Y 2 ] + Ξ If Y (conic) convex: can be achieved by duality theory!

18 Example problem: Linear Decision Rules three factories produce single good, one warehouse limited per-period production and storage capacities demand uniformly distributed among known nominal demand ( [ ]) nominal demand seasonal: d t = 1, sin π(t 1) 12

19 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

20 Bounding the Optimality Gap 2 So far: obtained upper bound on optimal value via restriction from general to linear decision rules Idea: obtain lower bound on optimal value to bound suboptimality 2 Kuhn et al., Math. Programming, 2010.

21 Bounding the Optimality Gap Step 1: dualise stochastic program with general decision rules Result: dual stochastic program with general decision rules

22 Bounding the Optimality Gap Step 2: restrict dual stochastic program to linear decision rules Result: dual stochastic program with linear decision rules allows us to bound incurred suboptimality

23 Example problem: Example Problem Revisited three factories produce single good, one warehouse limited per-period production and storage capacities demand uniformly distributed among known nominal demand ( [ ]) nominal demand seasonal: d t = 1, sin π(t 1) 12

24 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

25 Piecewise Linear Decision Rules 3 Linear decision rules can fail: minimise y E[y(ξ)] subject to y(ξ) ξ 1 P-a.s. where ξ U [ 1, 1] k. Remedy: use piecewise linear decision rules instead 3 Goh & Sim, Oper. Research, 2010; Georghiou et al., Optimization Online, 2010.

26 Piecewise Linear Decision Rules Piecewise linear decision rule linear decision rule in lifted space: y(ξ) y( ξ) ξ ξ ξ Ξ { } ) y(ξ) = y 0 + y 1 (min ξ, ξ ξ { + y 2 (max ξ, ξ } ξ ) } y 1 ξ Ξ ξ ξ ξ y( ξ) = y 0 + y 1 ξ1 + y 2 ξ2 { where ξ = min ξ, ξ } ξ { } max ξ, ξ ξ y 2 y 0 y 0 y 1 y 2

27 Piecewise Linear Decision Rules Example problem: capacity expansion of a power grid 10 regions with uncertain demand 5 power plants with known capacity, uncertain operating costs 24 transmission lines with known capacity goal: meet demand at lowest expected costs, via capacity expansion plan (here-and-now) plant operating policies (wait-and-see)

28 Outline 1 Deterministic Optimisation 2 Stochastic Programming 3 Scenario-Based Stochastic Programming 4 Scenario-Free Stochastic Programming Linear Decision Rules Bounding the Optimality Gap Piecewise Linear Decision Rules 5 Case Study: Capacity Expansion in Power Systems

29 Capacity Expansion in Power Systems Multiple time scales: Decades Years Time to delivery construct power plants & transmission lines Days schedule inflexible plants Hours Milliseconds schedule flexible plants power flows Nonrenewable energy sources: Natural gas, coal-fired Renewable energy sources: Solar and wind power

30 Capacity Expansion in Power Systems Four-stage stochastic program: Objective: minimise investment costs + expected operating costs over next 30 years Decades Years Time to delivery Days Hours Milliseconds investment decisions operating decisions (long lead) operating decisions (short lead) power flows forecasts of: fuel prices demand, wind, irradiation realisations of: fuel prices demand, wind, irradiation line failures

31 Optimisation Model minimise subject to c nu n + ( ) d mv m +E γ ng n n N c m M c n N g n F l -measurable n N l g n F s-measurable n N s f m F-measurable m M u n {0, 1}, v m {0, 1} n N, m M u n = 1 n N e v m = 1 m M e 0 g n g n u n n N g n ζ n n N r f m ϕ mf mv m m M g n f m + f m δ k k K n N(k) m M (k) m M +(k) P-a.s.

32 Existing Infrastructure Power system: 265 buses, 429 transmission lines

33 Extension Options Possible additions: 217 power plants, 81 transmission lines Existing Power Plants Candidate Power Plants

34 Solution time: 2.5 hours Results

35 Bibliography 1 A. Ben-Tal, A. Goryashko, E. Guslitzer, A. Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical Programming 99(2): , D. Kuhn, W. Wiesemann, A. Georghiou. Primal and dual linear decision rules in stochastic and robust optimization. Mathematical Programming, available via Online First, J. Goh, M. Sim. Distributionally Robust Optimization and its Tractable Approximations. Operations Research 58(4): , A. Georghiou, W. Wiesemann, D. Kuhn. Generalized Decision Rule Approximations for Stochastic Programming via Liftings. Optimization Online, 2010.